Re: [Phys-l] power, energy flow, vectors, tensors, et cetera.

[John Denker <jsd@av8n.com>]

On 07/28/2011 04:34 AM, David Bowman wrote:
> Confusing a power with time rate of energy density change is sort of
> like confusing a heat capacity with a specific heat, or a resistance
> with a resitivity.  One is defined for a specified finite sized
> system or object, and the other is defined differentially for the
> stuff or material out of which the particular system is made.

That's entirely true, and often important in practice.

We agree that I should have explained what I was doing.  I actually
composed an explanation but forgot to write it down.  It goes like
this:

 *) If a conservation law for some intensive quantity holds pointwise
  at every point in some region, then the corresponding law must hold
  for the corresponding extensive quantity.

 *) Conversely, if a conservation law holds for some extensive
  quantity in a certain region /and every subregion/ then the
  corresponding law must hold for the corresponding intensive
  quantity, pointwise, at every point in the region.

  This assumes the intensive quantity exists, i.e. that the
  extensive quantity is differentiable.

We speak of the "differential form of the law" and the "integral
form of the law".

There are times when a question that was asked in terms of an
extensive quantity is best answered in terms of the corresponding
intensive quantity.  For example:

 -- You can find tables of specific heat capacity for various
  materials more easily than you can find tables of the extensive heat
  capacity.

 -- You can google for "stress energy tensor" more easily than
  you can google for the corresponding extensive quantity.  If
  the latter even has a name, I don't know it.


FWIW note that the very name "stress energy tensor" is inconsistent
unto itself.  Stress is intensive, whereas energy is extensive.  That
doesn't make it right, but it is the conventional name.  Once again we
get to choose between what is conventional and what is right.